Magnetisation dynamics at Magnetisation dynamics at different - PowerPoint PPT Presentation

Magnetisation dynamics at Magnetisation dynamics at different timescales: different timescales: dissipation and thermal dissipation and thermal processes. processes. Numerical modelling methodology. Numerical modelling methodology. O.Chubykalo-Fesenko O.Chubykalo-Fesenko Instituto de Ciencia de Instituto de Ciencia de Materiales de Madrid, Spain Materiales de Madrid, Spain

Objective: large-scale modelling of complex ferromagnetic materials Patterned Lithographed Fe antidots FePt magnetic media Fe elongated nanoparticles prepared by extrusion CoCrPt magnetic Self-organized Co nanoparticles recording media FePt nanoparticles Prepared by laser ablation Very soft magnetic material: SmCo for hard magnets FePt nanoparticles Finemet

Objective: modelling of Objective: modelling of technological processes technological processes Conventional magnetic recording Heat-assisted magnetic recording Ultra-fast (fs) Kerr dynamics All-optical magnetic recording

Introduction Introduction • Magnetic system is not isolated, the magnetisation change can occur at any timescale. • Magnetism is a quantum phenomena. • Ab-initio calculations, although rapidly developing, at the present state of art are not capable to calculate magnetisation dynamics in complex materials at arbitrary timescale and temperature. • At larger spatial scale, relatively large magnetisation volumes (10nm) can be considered as classical particles.

The exchange term: micromagnetics The exchange term: micromagnetics versus spin models versus spin models •Micromagnetics calculates the magnetostatic fields exactly but which is forced to introduce an approximation to the exchange valid only for long-wavelength magnetisation fluctuations. •The exchange energy is essentially short ranged and involves a summation of the nearest neighbours. Assuming a slowly spatially varying magnetisation the exchange energy can be written E exch =  W e dv, with W e = A(  m ) 2 with (  m ) 2 = (  m x ) 2 + (  m y ) 2 + (  m z ) 2 The material constant A = JS 2 /a for a simple cubic lattice with lattice constant a. A includes all the atomic level interactions within the micromagnetic formalism.     exch E J S . S i ij i j •Atomistic models are discrete and use the Heisenberg  j i form of exchange

Micromagnetic models of Micromagnetic models of nanostructured materials nanostructured materials Models need nanostructure and micromagnetic parameters from experiment

Natural Natural magnetisation magnetisation dynamics: dynamics: 100 pico- 100 100 pico- 100 nano-second nano-second timescale timescale

Outline for today: 100ps- Outline for today: 100ps- 100ns (natural) dynamics 100ns (natural) dynamics • Non-thermal dynamics: • Ferromagnetic resonance • Basic dynamical equation: the Landau-Lifshitz- Gilbert • The problem of magnetic damping (  ): main processes • Thermal dynamics: • Principles of the Langevin dynamics. • Modelling of thermal spinwaves • The Landau-Lifshitz-Bloch micromagnetics for dynamics close to Tc

Ferromagnetic resonance(FMR): Ferromagnetic resonance(FMR): (Arkadiev, 1911; Kittel, 1947) (Arkadiev, 1911; Kittel, 1947)     A ferromagnetic body under        ( ) ( ) H N N M H N N M x z y z applied field has a maximum The absorption peak contains information absorption in frequencies: about anisotropy field. Torque on magnetisation The absorption line width contains Information on damping processes

Ferromagnetic resonance Ferromagnetic resonance • The experiment is normally performed in almost saturated conditions. • The absorption peak contains information about anisotropy field. • The linewidth contains information about dissipation processes.

FMR tecniques as a probe of FMR tecniques as a probe of magnetisation dynamics magnetisation dynamics Courtesy of G.Kakazei et al

The Landau-Lifshitz (LL) and the The Landau-Lifshitz (LL) and the Landau-Lifshitz-Gilbert (LLG) equations Landau-Lifshitz-Gilbert (LLG) equations of motion of motion (for magnetization vector): LL equation  Landau-Lifshitz damping, 1935   '      d M              ' M H LL 0 M M H 0 dt M s Gilbert equation (physically more reasonable Gilbert damping, 1955 for large damping)         d M   d M    0    M H G M   dt M dt   s How the Gilbert equation could be transformed into the LL equation LLG equation           The LL eq. is equivalent to G equation d M   d M          M M M H G M M   0 with substitutions dt M dt   s         d M d M            ' 0 , G   M M M M dt dt   0 LLG     2 2 1 1 G G   2 0 M s

Convenient form of the LLG Convenient form of the LLG equation: : equation  M         m , E E / 2 KV      2 / H K t 1 M G s     Contains all contributions: anisotropy, H E   , h int  Exchange, magnetostatic, Zeeman,  H m K depends on M 2 K  H - Anisotropy field K M s        d m           m h m m h  d

The Bloch-Bloembinger The Bloch-Bloembinger damping: damping:      d M   1        M H M Transverse relaxation 0 X , Y X , Y dt T   2 X , Y    d M     1         ( ) M H M M Longitudinal relaxation 0 Z s Z dt T   1 Z

The problem of damping: : The problem of damping • Different relaxation processes:  Magnon-magnon scattering  Magnon-electron interactions (especially in metals)  Phonon-magnon interactions (magnetostriction)  Impurities  Extrinsic factors (grain boundary, surface roughness, etc.)  Temperature disorder

Theory of magnetic Theory of magnetic damping constant   ) )   damping constant Uniform motion Spin waves Electron system Dissipation in lattice Impurities Surrounding body

Ferromagnets and their spin excitations         H J S S g S H Heisenberg Heisenberg i j B i Hamiltonian Hamiltonian  i , j i g   B M S i V i 2. Spin Waves 1. Uniform precession (ellipsoid)  M     ( M H ); eff  t    H . 0 eff H0   k    2 ( ) ( 0 ) Ak E More generally:  /  k=2  /  k=2 M M        H E / M . eff tot k Courtesy of K.Guslienko

Kittel formula for spinwaves Kittel formula for spinwaves dispersion relation: dispersion relation: Anisotropic single crystal ferromagnet: Angle between M and k 2                 2 2 2 sin H H Ak H H Ak 2 M    0 A 0 A s k   Exchange Magnetostatic Applied Anisotropy interaction interaction field field     k

Magnons and their Magnons and their interactions: interactions: • Classical spinwaves correspond to quasiparticles called magnons. • Homogeneous magnetisation (FMR mode) corresponds to magnon with k=0. • Linear normal modes (magnons) do not interact. Nonlinear processes correspond to magnon- magnon interactions. Magnon decay Two magnon merging magnon scattering 1 2 3 1 1 3 3 2 4 2 These interactions define kinetic effects (e.x. heat conductivity) and width and shape of the FMR line and magnon lifetime

Nonlinear phenomena: Suhl Nonlinear phenomena: Suhl instabilities. instabilities. • For large excitation power - FMR saturation occurs • If the density of magnons gets higher than critical value – the homogeneous oscillations become unstable         ( k ) ( k ) 2 ( k ) 0 k k=0 The occurrence of the instability depends on the system geometry and is governed by the applied field. -k         ( ) ( ) ( ) 2 k k 2 k Second condition, 0 more easy to meet

Inherent relaxation processes Inherent relaxation processes (via spin-wave instabilities) (via spin-wave instabilities) • Even without external dissipation it is possible to reach magnetisation reversal via spin-wave instabilities.

Main non-inherent relaxation Main non-inherent relaxation processes: processes: o Direct spin-lattice relaxation due to nonuniformities  Heterogeneity of composition  Polycrystalline structure (grain coundaries, etc.)  Nonuniform stresses, dislocations  Geometrical roughness: pores, surfaces etc. o Indirect spin-lattice relaxation  Via ions with strong spin-orbital coupling  Via charge carriers